A332354 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of triangular cells in the partition, for m >= n >= 1.
1, 2, 5, 3, 8, 15, 4, 11, 22, 33, 5, 14, 31, 48, 71, 6, 17, 40, 63, 94, 125, 7, 20, 51, 82, 125, 168, 227, 8, 23, 62, 101, 156, 211, 286, 361, 9, 26, 75, 124, 193, 262, 357, 452, 567, 10, 29, 88, 147, 230, 313, 428, 543, 682, 821, 11, 32, 103, 174, 275, 376, 517, 658, 829, 1000, 1219
Offset: 1
Examples
Triangle begins: 1, 2, 5, 3, 8, 15, 4, 11, 22, 33, 5, 14, 31, 48, 71, 6, 17, 40, 63, 94, 125, 7, 20, 51, 82, 125, 168, 227, 8, 23, 62, 101, 156, 211, 286, 361, 9, 26, 75, 124, 193, 262, 357, 452, 567, 10, 29, 88, 147, 230, 313, 428, 543, 682, 821, ...
Links
- M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. See Theorem 13.
- N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]
Programs
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Maple
# VR(m,n,q) is f_q(m,n) from the Alekseyev et al. reference. VR := proc(m,n,q) local a,i,j; a:=0; for i from -m+1 to m-1 do for j from -n+1 to n-1 do if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end; ct3 := proc(m,n) local i; global VR; if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end; for m from 1 to 12 do lprint([seq(ct3(m,n),n=1..m)]); od: